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Tools and Mathematics pp 219–256Cite as

Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France

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Part of the book series: Mathematics Education Library ((MELI,volume 110))

Abstract

This chapter analyses specificities of the French field of ‘didactics of mathematics’, questioning its reasons, tracing the geneses of concepts related to artefacts and following influences on, and interactions with the international communities of research. This complex genesis is traced in four sections: a first section on the roots of the didactics of mathematics in France, a second section on two founding theoretical frameworks (the theory of didactical situations of Brousseau, and the theory of conceptual fields of Vergnaud), a third section on the anthropological approach of Chevallard, a fourth focusing on specific approaches dedicated to artefacts and resources in mathematics education. Beyond historical and cultural specificities, the chapter aims to evidence the potential of interactions between teachers and researchers, as well as interactions between researchers in mathematics and mathematics education for improving our understanding of learning and teaching issues in mathematics.

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Notes

  1. 1.

    I have chosen to give sometimes long quotations, keeping the words and the language—sometimes in French—of these authors, in order to allow the reader to have a direct contact with their works.

  2. 2.

    I would like to thank Janine Rogalski and Rudolf Straesser for their comments and advices.

  3. 3.

    CREM: Commission de réflexion sur l’enseignement des mathématiques.

  4. 4.

    The digital copies of the journal can be found at http://www.unige.ch/math/EnsMath/EM_fr/welcome.html.

  5. 5.

    Informations from the ICMI website http://www.mathunion.org/icmi/icmi/executive-committee/past-executive-committees/.

  6. 6.

    The Musée pédagogique is the forerunner of the National Institute for Pedagogical Research, which became in 2010 the French Institute of Education.

  7. 7.

    Note that this did not come, for the 1967 reform, from the whole Bourbaki group, but only from its members interested in changing the teaching at a secondary level. Houzel (2004, p. 57) wrote, on this particular question: “In the late 60s, a reform movement in secondary mathematics education has been launched in most countries and this movement has unfortunately claimed Bourbaki. From it, came what was called the ‘new math’, whose harmfulness is no longer in doubt. But it is unfair to shift the burden to Bourbaki, whose only fault was to ignore the problem of Dieudonné propaganda rather dangerous to teachers (our translation)”

  8. 8.

    It is indeed significant that the first ICMI study, 1985) was dedicated to the Influence of Computers and Informatics on the Mathematics and its Teaching (Cornu & Ralston, 1992).

  9. 9.

    In 1910 was also created the association « L’école émancipée » [The emancipated school], gathering pedagogical activist teachers and revolutionary syndicalists.

  10. 10.

    The democratic goal of the French republic was to move from a schooling system founded on ‘orders’ (schools for people vs. schools for upper classes) to a schooling system founded on ‘levels’ (primary level vs. secondary level), i.e. a same school for each student: ‘l’école unique’. Several successive laws (1959, 1963, 1966) constituted a progress towards this objective, never fully achieved.

  11. 11.

    http://faculty.washington.edu/warfield/guy-brousseau.com

  12. 12.

    Extract of the third part of the preliminary report of the commission Lichnerowicz, published in the « Bulletin of APMEP », no. 258.

  13. 13.

    Artigue and Douady (p. 85) underline that “the expression didactics of mathematics has been introduced by Klein in 1910”.

  14. 14.

    French national centre for scientific research.

  15. 15.

    This growth mirrors the growth of mathematics education as a field noted in Sect. 7.2.

  16. 16.

    I have chosen to give, in this section, recent references to the work of Brousseau and Vergnaud, offering a more synthetic view on their work, but it has to be clear that the foundations of their theories come from the 1970s.

  17. 17.

    http://guy-brousseau.com/le-corem/

  18. 18.

    In a recent paper, Brousseau (2012) came back to the “psychological and didactical roots” of his theory, acknowledging the importance, among other researchers, of Greco and Piaget. He has developed a website http://faculty.washington.edu/warfield/guy-brousseau.com where could be found various elements grounding his approach.

  19. 19.

    For distinguishing Brousseau’s notion from the non-specific, standard uses of the word ‘situation’, she chose to capitalize this as soon as it is used in the frame of the Theory of didactical situations. I have not retained here such a convention: it is enough to consider that, in this section related to Brousseau’s theory, the word ‘situation’ is used with respect to this theory.

  20. 20.

    In French in the text/

  21. 21.

    In French in the text.

  22. 22.

    This proposition results also of interactions with a French didactician, Alain Mercier I want to thank here.

  23. 23.

    More on Vergnaud theory and publications can be found on the French mathematics didactics website: http://www.ardm.eu/contenu/gérard-vergnaud-english.

  24. 24.

    Chevallard offers, on his website, most of his publications: http://yves.chevallard.free.fr.

  25. 25.

    Translation resulting from interactions between Marianna Bosch and John Monaghan.

  26. 26.

    For a discussion on this point, see Bosch and Chevallard (1999, p. 29).

  27. 27.

    Contrary to what is said in the previous quote of Guin and Trouche (1999), corresponding to a previous step of the genesis instrumental approach of didactics, an artefact is not necessarily material. It can be also symbolic, as an algorithm, or a language. Its structural characteristic is to be a result of human activity, and to be potentially engaged in new activity.

  28. 28.

    For example, Hembree and Dessart (1986), see Sect. 13.2.

  29. 29.

    The notion of ‘problematic’ comes from the French ‘problématique’, well defined by Edward Said: ‘The idea of beginning, indeed the act of beginning [a research], necessarily involves an act of delimitation by which something is cut out of a great mass of material, separated from the mass, and made to stand for, as well as be, a starting point, a beginning; […] such notion of inaugural delimitation is Louis Althusser’s idea of the problematic, […] is something given rise to by analysis’ (Said 1978, p. 24).

References

  • Artigue, L. (1997). Rapports entre dimensions technique et conceptuelle dans l’activité mathématique avec des systèmes de mathématiques symboliques. In Actes de l’Université d’été « Des outils informatiques dans la classe aux calculatrices symboliques et géométriques : quelles perspectives pour l’enseignement des mathématiques ? » (pp. 19–40). Rennes, France: IREM.

    Google Scholar 

  • Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. doi:10.1016/s0277-5395-00-00166-7.

    Article  Google Scholar 

  • Artigue, M., & Douady, R. (1986). La didactique des mathématiques en France—Emergence d'un champ scientifique. Revue française de pédagogie, 76, 69–88. Retrieved September 15, 2014, from http://www.persee.fr/web/revues/home/prescript/article/rfp_0556-7807_1986_num_76_1_1503

  • Artigue, M., Gras, R., Laborde, C., & Tavignot, P. (Eds.). (1994). Vingt ans de didactique des mathématiques en France. Hommage à Guy Brousseau et Gérard Vergnaud. Grenoble, France: La pensée sauvage éditions.

    Google Scholar 

  • Balacheff, N. (1996). Advanced educational technology: Knowledge revisited. In T. T. Liao (Ed.), Advanced educational technology: Research issues and future potential (pp. 1–20). Berlin: Springer.

    Chapter  Google Scholar 

  • Barbazo, E. (2010). L’APMEP, un acteur politique, scientifique, pédagogique de l’enseignement secondaire mathématique du 20e siècle en France [APMEP, a political, scientific and pedagogic actor in 20th century French secondary mathematics teaching]. PhD. Paris, France: EHESS.

    Google Scholar 

  • Bass, H. (2008). Moments in the life of ICMI. In M. Menghini, F. Furinghetti, L. Giacardi, & F. Arzarello (Eds.), The first century of the international commission on mathematical instruction (1908-2008). Reflecting and shaping the world of mathematics education (pp. 9–24). Rome: Instituto della Enciclopedia Italiana.

    Google Scholar 

  • Belhoste, B. (1990). L’enseignement secondaire français et les sciences au début du XXe siècle. La réforme de 1902 des plans d’études et des programmes. Revue d’histoire des sciences, 43(4), 371–400. Retrieved from http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1990_num_43_4_4502

    Google Scholar 

  • Biehler, R., Scholz, R. W., Sträßer, R., & Winkelmann, B. (Eds.). (1994). Didactics of mathematics as a scientific discipline. Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Billeter, J.-F. (2002). Leçons sur Tchouang-Tseu. Paris: Editions Allia.

    Google Scholar 

  • Borel, E. (1904). Les exercices pratiques de mathématiques dans l’enseignement secondaire [Practical exercises in mathematics for secondary instruction]. Revue générale des sciences pures et appliquées, 14, 431–440.

    Google Scholar 

  • Bosch, M., & Chevallard, Y. (1999). La sensibilité de l’activité mathématique aux ostensifs. Objet d’étude et problématique. Recherches en didactique des mathématiques, 19(1), 79–124.

    Google Scholar 

  • Bourbaki, (1962). L’architecture des mathématiques [The architecture of mathematics]. In François Le Lionnais (Ed.), Les grands courants de la pensée mathématique (2nd ed., pp. 35–47). Paris: Albert Blanchard.

    Google Scholar 

  • Bronckart, J.-P., & Schurmans, M.-N. (1999). Pierre Bourdieu—Jean Piaget : habitus, schèmes et construction du psychologique. In B. Lahire (Ed.), Le travail sociologique de Pierre Bourdieu. Dettes et critiques. Paris: La Découverte.

    Google Scholar 

  • Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Brousseau, G. (2012). Des dispositifs Piagétiens… aux situations didactiques [From Piagetian experimental designs … to didactical situations]. Education et didactique, 6(2), 103–129. Retrieved from http://educationdidactique.revues.org/1475

    Google Scholar 

  • Brousseau, G., Brousseau, N., & Warfield, G. (2014). Teaching fractions through situations: A fundamental experiment. Dordrecht, The Netherlands: Springer.

    Book  Google Scholar 

  • Chevallard, Y. (1988, August). On didactic transposition theory: Some introductory notes. In International Symposium on Research and Development in Mathematics, Bratislava, Czechoslavakia. Retrieved November 14, from http://yves.chevallard.free.fr/spip/spip/IMG/pdf/On_Didactic_Transposition_Theory.pdf

  • Chevallard, Y. (1995). Les outils sémiotiques du travail mathématique. « petit x », 42, 33–57.

    Google Scholar 

  • Chevallard, Y. (1997). Familière et problématique, la figure du professeur. Recherches en didactique des mathématiques, 17(3), 17–54.

    Google Scholar 

  • Chevallard, Y. (2005). Steps towards a new epistemology in mathematics education. In M. Bosch (ed.), Proceedings of the Fourth European Conference on Research on Mathematics Education (pp. 21–30). FUNDEMI IQS—Universitat Ramon Llull. Retrieved from http://ermeweb.free.fr/CERME4

  • Cornu, B., & Ralston, A. (Eds.). (1992). The influence of computers and informatics on mathematics and its teaching. ICMI Study Conference Held in Strasbourg, France, March 1985. Second edition published by UNESCO. Retrieved from http://unesdoc.unesco.org/images/0009/000937/093772eo.pdf

  • Dienes, Z. P. (1970). Les six étapes du processus d’apprentissage en mathématiques. Paris: OCDL.

    Google Scholar 

  • Douady, R. (1987). Jeux de cadres et dialectique outil objet. Recherches en didactique des mathématiques, 7(2), 5–31.

    Google Scholar 

  • Douglas, M. (1986). How institutions think? Syracuse, NY: Syracuse University Press.

    Google Scholar 

  • Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75, 213–234.

    Article  Google Scholar 

  • Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2012). One episode, two lenses. A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educational Studies in Mathematics, 82, 23–49.

    Article  Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. Retrieved from http://www.cimm.ucr.ac.cr/ojs/index.php/eudoxus/article/viewFile/162/297

    Google Scholar 

  • Engeström, Y., Miettinen, R., & Punamaki, R.-L. (1999). Perspectives on activity theory. Cambridge, England: Cambridge University Press.

    Book  Google Scholar 

  • Gispert, H. (2014). Mathematics education in France, 1900-1980. In A. Karp & G. Schubring (Eds.), Handbook on the history of mathematics education (pp. 229–240). New York: Springer.

    Chapter  Google Scholar 

  • Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3), 199–218.

    Article  Google Scholar 

  • Gueudet, G., & Vandebrouck, F. (2011). Technologies et évolution des pratiques enseignantes : études de cas et éclairages théoriques. Recherches en didactique des mathématiques, 31(3), 271–313.

    Google Scholar 

  • Guin, D., Ruthven, K., & Trouche, L. (Eds.). (2005). The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. New York: Springer.

    Google Scholar 

  • Guin, D., & Trouche, L. (1999). The complex process of converting tools into mathematical instruments. The case of calculators. The International Journal of Computers for Mathematical Learning, 3(3), 195–227. Retrieved from http://link.springer.com/article/10.1023/A:1009892720043#page-1

  • Hembree, R., & Dessart, D. J. (1986). Effects of hand-held calculators in precollege mathematics education: A meta-analysis. Journal for Research in Mathematics Education, 17(2), 83–99.

    Article  Google Scholar 

  • Houzel, C. (2004). Le rôle de Bourbaki dans les mathématiques du vingtième siècle. Gazette des Mathématiciens, 100, 53–63.

    Google Scholar 

  • Hoyles, C., & Lagrange, J.-B. (2006). The seventeenth ICMI study. Technology revisited. Discussion document. Retrieved December 1, from http://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/DiscussionDocs/DD_icmiStudy17_02.pdf

  • Hoyles, C., Noss, R., & Kent, P. (2004). On the integration of digital technologies into mathematical classrooms. International Journal of Computers for Mathematical Learning, 9, 309–326.

    Article  Google Scholar 

  • Kahane, J.-P. (dir.). (2002). L’enseignement des sciences mathématiques. Rapport au ministre. Commission de réflexion sur l’enseignement des mathématiques. Odile Jacob.

    Google Scholar 

  • Kahane, J.-P. (2006). Cooperation and competition as a challenge in and beyond the classroom. In ICMI study no. 16 conference. Retrieved from http://www.mathunion.org/fileadmin/ICMI/files/Conferences/ICMI_studies/Study16/icmis16pkahane_03.pdf

  • Kilpatrick, J. (1994). Vingt ans de didactique française depuis les USA. In M. Artigue, R. Gras, C. Laborde, & P. Tavignot (Eds.), Vingt ans de didactique des mathématiques en France. Hommage à Guy Brousseau et Gérard Vergnaud (pp. 84–96). Grenoble, France: La pensée sauvage éditions.

    Google Scholar 

  • Lagrange, J.-B., Artigue, M., Laborde, C., & Trouche, L. (2003). Technology and mathematics education: A multidimensional study of the evolution of research and innovation. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 239–271). Dordrecht, The Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Lévy-Strauss, C. (1949). Les structures élémentaires de la parenté. Paris: PUF.

    Google Scholar 

  • Linhart, R. (1978). L’établi. Paris: Éditions de minuit.

    Google Scholar 

  • Margolinas, C. (2002). Situations, milieux, connaissances : analyse de l’activité du professeur. In J.-L. Dorier, M. Artaud, M. Artigue, R. Berthelot, & R. Floris (Eds.), Actes de la 11 ème Ecole d’Eté de Didactique des Mathématiques (pp. 141–156). Grenoble, France: La pensée sauvage.

    Google Scholar 

  • Mariotti, M. (2002). The influence of technological advances on students’ mathematics learning. In L. English, M. G. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 695–723). Hillsdale, NJ: Lawrence Erbaum Associates.

    Google Scholar 

  • Maschietto, M., & Trouche, L. (2010). Mathematics learning and tools from theoretical, historical and practical points of view: The productive notion of mathematics laboratories. ZDM, International Journal on Mathematics Education, 42(1), 33–47.

    Article  Google Scholar 

  • Mauss, M. (1966). The gift: Forms and functions of exchange in archaic societies. London: Cohen & West (original edition).

    Google Scholar 

  • Menghini, M., Furinghetti, F., Giacardi, L., & Arzarello, F. (Eds.). (2008). The first century of the international commission on mathematical instruction (1908-2008). Reflecting and shaping the world of mathematics education (pp. 9–24). Rome: Instituto della Enciclopedia Italiana.

    Google Scholar 

  • Monaghan, J. (2001). Teachers’ classroom interactions in ICT-based mathematics lessons. In M. van den Heuvel (Ed.), Proceedings of the 25th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 383–390). Utrecht, The Netherlands.

    Google Scholar 

  • Piaget, J., Beth, E. W., Dieudonné, J., Lichnerowicz, A., Choquet, G., & Gattegno, C. (1955). L’enseignement des mathématiques. Neuchâtel, Switzerland: Delachaux & Niestlé.

    Google Scholar 

  • Poincaré, H. (1904). Les définitions générales en mathématiques. L’Enseignement mathématique, 6, 257–283.

    Google Scholar 

  • Prediger, S., Arzarello, F., Bosch, M., & Lenfant, A. (Eds.). (2008). Comparing, combining, coordinating—Networking strategies for connecting theoretical approaches. ZDM, The International Journal on Mathematics Education, 40(2), pp. 163–327.

    Google Scholar 

  • Rabardel, P. (2000). Eléments pour une approche instrumentale en didactique des mathématiques. In M. Bailleul (Ed.), Actes de l’Ecole d’été de didactique des mathématiques, IUFM de Caen, pp. 203–213.

    Google Scholar 

  • Robert, A., & Rogalski, J. (2002). Le système complexe et cohérent des pratiques des enseignants de mathématiques : une double approche. Canadian Journal of Science, Mathematics and Technology Education (La revue canadienne de l’enseignement des sciences, des mathématiques et des technologies), 2(4), 505–528.

    Article  Google Scholar 

  • Rouchier, A. (1994). Naissance et développement de la didactique des mathématiques. In M. Artigue, R. Gras, C. Laborde, & P. Tavignot (Eds.), Vingt ans de didactique des mathématiques en France. Hommage à Guy Brousseau et Gérard Vergnaud (pp. 148–160). Grenoble, France: La pensée sauvage editions.

    Google Scholar 

  • Said, E. W. (1978). Orientalism. London: Routledge & Kegan.

    Google Scholar 

  • Schubring, G. (2010). Historical comments on the use of technology and devices in ICMEs and ICMI. ZDM, The International Journal on Mathematics Education, 42(1), 5–9.

    Article  Google Scholar 

  • Strässer, R. (1994). A propos de la transposition franco-allemande en didactique des mathématiques. In M. Artigue, R. Gras, C. Laborde, & P. Tavignot (Eds.), Vingt ans de didactique des mathématiques en France. Hommage à Guy Brousseau et Gérard Vergnaud (pp. 161–176). Grenoble, France: La pensée sauvage editions.

    Google Scholar 

  • Sträßer, R. (2009). Instruments for learning and teaching mathematics an attempt to theorize about the role of textbooks, computers and other artefacts to teach and learn mathematics. In Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 67–81).

    Google Scholar 

  • Sutherland, R., & Balacheff, N. (1999). Didactical complexity of computational environments for the learning of mathematics. International Journal of Computers for Mathematical Learning, 4(1), 1–26.

    Article  Google Scholar 

  • Trouche, L. (1997). A la recherche d’une méthode d’étude de “l’action instrumentée”. In Actes de l’Université d’été « Des outils informatiques dans la classe aux calculatrices symboliques et géométriques : quelles perspectives pour l’enseignement des mathématiques » (pp. 113–148). Rennes, France: IREM.

    Google Scholar 

  • Trouche, L. (2009). Penser la gestion didactique des artefacts pour faire et faire faire des mathématiques : histoire d’un cheminement intellectuel. L’Educateur, 0309, 35–38.

    Google Scholar 

  • Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307.

    Article  Google Scholar 

  • Trouche, L., & Drijvers, P. (2014). Webbing and orchestration. Two interrelated views on digital tools in mathematics education. Teaching Mathematics and Its Applications, 33, 193–209. doi:10.1093/teamat/hru014.

    Article  Google Scholar 

  • Vergnaud, G. (2000). Lev Vygotski. Pédagogue et penseur de notre temps. Paris: Hachette.

    Google Scholar 

  • Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52, 83–94.

    Article  Google Scholar 

  • Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.

    Article  Google Scholar 

  • Warfield, V. M. (2014). Introduction to didactics. SpringerBriefs in education 30. New York: Springer.

    Google Scholar 

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    Luc Trouche

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Trouche, L. (2016). Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France. In: Tools and Mathematics. Mathematics Education Library, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-02396-0_10

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